Chapter 11
11.1
(a)
3
1 2 4
3
1 2 4
3
1 2 4
33
1 1 2 2 4 4
33
1 1 2 2 4 4
33
1 1 2 2 4 4
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
10 0 0 0 0 0 0 0
60 0 0 0
0 0 0 0
0 0 0 0
V
By Equations (11.2.4) to (11.2.8)
0
12
0
12
0
12
00
1
0
12
6V =
00
12
00
12
00
12
000
1
= (1) (– 1)2
00
2
00
2
0 0 0
+ 2 (– 1)3
00
1 2
00
1 2
0 0
11
= 8
000000 00 0 0
44
0000 0000 0 0
44
100 000000 0 0
44
80 0 0 0 0 0 0 0
4 4 4 4
0 0 0 0 0 0 0 0
4 4 4 4
0000000 0
4 4 4 4
Problem 11-1: ‘B’ matrix for tetrahedral solid element
(b)
x1 = 1 y1 = 0 z1 = 0
x2 = 0 y2 = 0 z2 = 2
44
1
yz
44
1
xz
44
1
xy
11
11
1
xy
1 1 1
x y z
1 1
11
y z
y z
1 1
11
x z
x z
11
1
xy
xy
11.2(a) Use Equation (11.2.18) and substitute [B] from 11.1 (a) and [D] from Equation (11.1.5) into
Equation (11.2.18)
[k] = [B]T [D] [B] V
1
2
3
4
5
6
7
8
9
10
11
12
1
0
0
0
0
0
0
0.5
0
0
–0.5
0
0
2
0
0
0
0
0.5
0
0
0
0
0
–0.5
0
[B] =
3
0
0
0.5
0
0
0
0
0
0
0
0
–0.5
4
0
0
0
0.5
0
0
0
0.5
0
–0.5
–0.5
0
5
0
0.5
0
0
0
0
0
0
0.5
0
–0.5
–0.5
6
0.5
0
0
0
0
0
0
0
0.5
–0.5
0
–0.5
v = 0.3 E = 30 106
1 – 0 0 0
0 1 – 0 0 0
0 0 1 – 0 0 0
1 – 2
0 0 0 0 0
2
(1 ) (1 – 2 ) 1 – 2
0 0 0 0 0
2
1 – 2
0 0 0 0 0 2
v v v
vv
v
v
E
vv
v
v
[k] = [B]T [D] [B] V
1
2
3
4
5
6
7
8
9
10
11
12
1
3.846
0
0
0
0
0
0
0
3.846
–3.846
0
–3.846
2
0
3.846
0
0
0
0
0
0
3.846
0
–3.846
–3.846
3
0
0
13.462
0
0
0
0
0
0
0
0
13.462
4
0
0
0
3.846
0
0
0
3.846
0
–
3.846
– 3.846
0
5
0
0
5.769
0
13.462
0
0
0
0
0
13.462
–5.769
[k] =
1
3x
6
0
0
0
0
0
0
0
0
0
0
0
0
7
0
0
5.769
0
5.769
0
13.462
0
0
13.462
–5.769
–5.769
8
0
0
0
3.846
0
0
0
3.846
0
–3.846
–3.846
0
9
3.846
3.846
0
0
0
0
0
0
7.692
–3.846
–3.846
–7.692
1
0
–3.846
0
–5.769
–3.846
–5.769
0
13.462
–3.846
–3.846
21.154
9.615
9.615
1
1
0
–3.846
–5.769
–3.846
13.462
0
0
–3.846
–3.846
3.846
21.154
9.615
1
2
–3.846
–3.846
13.462
0
0
0
0
0
–7.692
3.846
3.846
21.154
1 – 0 0 0
0 1 – 0 0 0
0 0 1 – 0 0 0
1 – 2
0 0 0 0 0
v v v
vv
v
v
E
11.3 (a) {
} = [B] {d}
x
y
z
xy
yz
zx
4
000000 00 0 0
4
0000 0000 0 0
44
4
100 000000 0 0
4
44
80 0 0 0 0 0 0 0
44
44
0 0 0 0 0 0 0 0
44
44
0000000 0
44
0.005
0.0
0.0
0.001
0.0
0.001
0.005
0.0
0.0
0.001
0.0
0.005
=
0.003
0.0
0.0025
0.001
–0.002
0.0005
in.
in.
3
3
3
2.885 10
2.885 10
6.730 10





458
10 0
1
40 0
1
10 0
1
25 25
1
1 = –
25 25
1
25 0
1
40 0
1
= – 375,
2 =
10 0
1
25 0
1
40 0
1
= 0
3 = –
10 0
1
25 25
1
40 0
1
= 750,
4 =
10 0
1
25 25
1
25 0
1
= – 375
10 0 0
1
25 25 0
1
25 25 0
18750 375 625 0 750 0 0 0 0 0 375 625 0
0 375 375 0 0 750 0 750 0 0 375 375
375 0 625 0 0 0 750 0 0 375 0 625
(b)
v =
1 1 1
2 2 2
3 3 3
4 4 4
1
11
61
1
x y z
x y z
x y z
x y z
1
yz
yz
11
1
xz
1
1
yz
11
1
1
2 2 2
x y z
22
1
yz
yz
22
1
xz
xz
22
1
xy
xy
0.125 0 0 0 0 0 0 0 0 0.125 0 0
0 0.05 0 0 0.2 0 0 0 0 0 0.15 0
0 0 0.05 0 0 0 0 0 0.2 0 0 0.15
0.05 0.125 0 0.2 0 0 0 0 0 0.15 0.125 0
0 0.05 0.05 0 0 0.2 0 0.2 0 0 0.15 0.
0.05 0 0.125 0 0 0 0.2 0 0
15
0.15 0 0.125
11.7
6
563568. 200321. 200321. 96153.8 240385. 0. 96153.8 0. 240385. 371261. 40064.1 40064.1
200321. 349893. 120192. 160256. 336538. 96153.8 0. 96153.8 144231 40064.1 82799.1 120192.
200321. 120192. 349893. 0. 144231
[k] 10
. 96153.8 160256. 96153.8 336538 40064.1 120192. 82799.1
96153.8 160256. 0. 192308. 0. 0. 0. 0. 0. 96153.8 160256. 0.
240385. 336538 144231. 0. 673077. 0. 0. 0. 288462. 240385. 336538. 144231.
0. 96153.8 96153.8 0. 0. 1923
 08. 0. 192308. 0. 0. 96153.8 96153.8
96153.8 0. 160256. 0. 0. 0. 192308. 0. 0. 96153.8 0 160256.
0. 96153.8 96153.8 0. 0. 192308. 0. 192308. 0. 0. 96153.8 96153.8
240385 144231. 336538. 0. 288462. 0. 0. 0. 673077. 240385. 144231

. 336538.
371261 40064.1 40064.1 96153.8 240385. 0. 96153.8 0. 240385. 563568. 200321. 200321.
40064.1 82799.1 120192. 160256. 336538. 96153.8 0. 96153.8 144231. 200321. 349393. 120192.
40064.1 120192. 82799.1 0. 144
231. 96153.8 160256. 96153.8 336538. 200321. 120192. 349393.
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


11.8 (a) {
} = [B] {d}
(see [B] from Problem 11.6 above)
u1 = 0 v1 = 0 w1 = 0
u2 = 0.00001 v2 = 0.00002 w2 = 0.00001
u3 = 0.00002 v3 = 0.00001 w3 = 0.000005